I have trouble proving this. Thanks for any help in advanced!

Prove the following:

tanxsinxtanx - sinx

tanx + sinx = tanxsinx

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- Dec 18th 2008, 03:10 PMArthurYantharTrigonomic Identities Help?
I have trouble proving this. Thanks for any help in advanced!

Prove the following:

__tanxsinx____tanx - sinx__

tanx + sinx = tanxsinx - Dec 18th 2008, 03:42 PMKrizalid
$\displaystyle \frac{\tan (x)\sin (x)}{\tan (x)+\sin (x)}=\frac{\tan (x)\sin (x)\left( \tan (x)-\sin (x) \right)}{\tan ^{2}(x)\sin ^{2}(x)}=\frac{\tan (x)-\sin (x)}{\tan (x)\sin (x)}.$

Observe that $\displaystyle \tan ^{2}(x)-\sin ^{2}(x)=\sin ^{2}(x)\left( \frac{1}{\cos ^{2}(x)}-1 \right)=\tan ^{2}(x)\sin ^{2}(x),$ and we're done. $\displaystyle \blacksquare$